Since AE is to EB as CF is to FD, therefore the square on AE is to the rectangle AE by EB as the square on CF is to the rectangle CF by FD. Therefore, alternately, the square on AE is to the square on CF as the rectangle AE by EB is to the rectangle CF by FD.

But the square on AE is commensurable with the square on CF, therefore the rectangle AE by EB is commensurable with the rectangle CF by FD.

Therefore if the rectangle AE by EB is rational, then the rectangle CF by FD is also rational, and for this reason CD is a first bimedial, but if medial, medial, and each of the straight lines AB and CD is a second bimedial. And for this reason CD is the same in order with AB.

Therefore, a straight line commensurable with a bimedial straight line is itself also bimedial and the same in order.